We say that $X$ is of geometric origin from a $d$-dimensional family over a field $k$ if there exist a normal $d$-dimensional variety $B$ over $k$, a point $b\in B^{(1)}$ of codimension one and a projective variety $Y$ over $K^{\prime}=k(B)$ such that

1. (i)

   there exists an isomorphism $\widehat{\mathcal{O}}_{B,b}\stackrel{{\scriptstyle\sim}}{{\to}}K^{\circ}$ of rings where $\widehat{\mathcal{O}}_{B,b}$ denotes the completion of the discrete valuation ring $R\coloneqq{\mathcal{O}}_{B,b}$,

2. (ii)

   an isomorphism $Y\otimes_{K^{\prime}}K\simeq X$ over $K$ with $K^{\prime}\to K$ induced by (i).

Usually, we read these isomorphisms as identifications. Moreover, if $L$ is a line bundle (resp. if $\theta$ is a closed $(1,1)$-form) on $X$, we say that $(X,L)$ (resp. $(X,\theta)$) is of geometric origin from a $d$-dimensional family over a field $k$ if the above conditions are satisfied and if we can also find a line bundle $L^{\prime}$ on $Y$ inducing $L$ by the base change $K/K^{\prime}$ (resp. a line bundle $\mathscr{L}_{R}$ on an $R$-model $\mathscr{X}_{R}$ of $Y$ inducing $\theta$ by the base change ${K^{\circ}}/R$).